net_matrix_lp: Estimate Migration Flows to Match Net Totals via Linear Programming

Description

Solves for an origin-destination flow matrix that satisfies directional net migration constraints while minimizing the total absolute deviation from a prior matrix. This method uses linear programming with split variables to minimize L1 error, optionally respecting a structural zero mask.

Usage

net_matrix_lp(net_tot, m, zero_mask = NULL, tol = 1e-06)

Value

A named list with components:

n: Estimated matrix of flows satisfying the net constraints.
it: Number of iterations (always 1 for LP method).
tol: Tolerance used for checking net flow balance.
value: Total L1 deviation from prior matrix m.
convergence: Logical indicator of successful solve.
message: Text summary of convergence status.

Arguments

net_tot: A numeric vector of net migration totals for each region. Must sum to zero.
m: A square numeric matrix providing prior flow estimates. Must have dimensions length(net_tot) × length(net_tot).
zero_mask: A logical matrix of the same dimensions as m, where TRUE indicates forbidden (structurally zero) flows. Defaults to disallowing diagonal flows.
tol: A numeric tolerance for checking that sum(net_tot) == 0. Default is 1e-6.

Details

This function uses lpSolve::lp() to solve a linear program. The estimated matrix minimizes the sum of absolute deviations from the prior matrix m, subject to directional net flow constraints: $$\sum_j x_{ji} - \sum_j x_{ij} = \text{net}_i$$ Structural zeros are enforced by the zero_mask. All flows are constrained to be non-negative.

Examples

Run this code

m <- matrix(c(0, 100, 30, 70,
              50,   0, 45,  5,
              60,  35,  0, 40,
              20,  25, 20,  0),
            nrow = 4, byrow = TRUE,
            dimnames = list(orig = LETTERS[1:4], dest = LETTERS[1:4]))
addmargins(m)
sum_region(m)

net <- c(30, 40, -15, -55)
result <- net_matrix_lp(net_tot = net, m = m)
result$n |>
  addmargins() |>
  round(2)
sum_region(result$n)